Lasso "Fill" Tool alternative for Texture Painting
Your question is for a Lasso Fill Tool, but the video is showing the combination of a Lasso Selection tool and a Bucket Fill tool. We have a Fill tool in the Paint tools, but we do not have a Lasso Selection tool.

The easiest work around for now is to use the Mask tool in the 3d view to paint a selection with a hard brush and then invert the mask, switch to the Fill tool and drop a color fill into the unmasked area. In the 2d Image Editor, you can use a hard brush to paint a selection with a solid contrasting color and then switch to the Fill tool and turn down the threshold level to only affect the colored pixels of the contrast color.

The 2d image editor solution is only good for solid fill and not for any extra painting inside after the fill as far as constraining the selection for paint. The 3d view Mask tool solution is very good and can be considered to be akin to a Layer Mask in Photoshop styled editors, except there are so many more ways to utilize more than the single UV mapping of your object with the Mask tool.

The creation of a Lasso Selection tool in Blender is a good topic for a proposal on Rightclickselect.com community page, and it may have been suggested there a few times now.